Solving and bounding Stein equations

Ferdinand Rapin and Yvik Swan, Representations of solutions to Stein equations and their derivatives (2026). arXiv:2609.17881 · Article PDF.

The method

For a density $p$ on $(\ell,u)$ , choose a nonvanishing weight $w$ and set $\gamma=(wp)'/p$ . The first-order Stein equation is

w(x)f'_h(x)+\gamma(x)f_h(x)=h(x)-\mathbb E[h(Z)].

With the appropriate boundary condition, integration gives the canonical solution

f_h(x)=\frac{\displaystyle\int_\ell^x\bigl(h(t)-\mathbb E[h(Z)]\bigr)p(t)\,dt}{w(x)p(x)}.

The method represents this solution and its derivatives through an array of signed kernels. Differentiation and integration identities move between derivative orders of the solution and of the test function. The choice of weight determines which correction terms cancel; in general these terms must be retained.

Controlling the test function and the absolute kernel integrals gives pointwise envelopes. Taking a supremum over $x$ gives uniform Stein factors; multiplying by a chosen weight gives weighted factors. Kernel signs can identify extremising tests and establish sharpness. Each bound depends on the test class, derivative order, moment assumptions, and boundary conditions. See the article for the representations and their hypotheses.

Companion papers and addendum

The following September 2026 preprints organise the bounds by test class: Kolmogorov, bounded measurable, Wasserstein, Fortet–Mourier, and integer-order Zolotarev. They distinguish exact envelopes, explicit upper bounds, and derivative orders for which the requested bound is unavailable.

Gaussian Stein envelopes and factors: a kernel calculus and a guide by test class (PDF)
Exact envelopes and sharp factors for the standard Gaussian equation.
Gamma Stein envelopes and factors: weighted and unweighted bounds by test class (PDF)
Bounds on the positive half-line, including endpoint behaviour and admissible weights.
F-distribution Stein envelopes and factors: ordinary and weighted bounds by test class (PDF)
F and scaled beta-prime equations, with moment thresholds for smooth tests.
Inverse Gamma Stein envelopes and factors: ordinary and weighted bounds by test class (PDF)
Inverse Gamma equations, tail behaviour, and weighted derivative bounds.
Student Stein envelopes and factors: ordinary and weighted bounds by test class (PDF)
Student equations with explicit moment requirements for each smoothness class.
Maxwell Stein envelopes and factors: ordinary and weighted bounds by test class (PDF)
The covariance-kernel normalisation and non-Pearson correction terms.
Symmetrized Maxwell Stein envelopes and factors: ordinary and weighted bounds by test class (PDF)
A canonical formulation valid across the origin, where the density vanishes.
Subbotin Stein envelopes and factors: the Laplace case (PDF)
The case $\beta=1$ , with the behaviour at the origin treated explicitly.
Subbotin Stein envelopes and factors: odd integer powers (PDF)
Odd integer powers $\beta\geq3$ , including regularity and higher-order obstructions.
Subbotin Stein envelopes and factors: noninteger powers (PDF)
Noninteger powers $\beta>1$ , with the applicable derivative orders specified.
Improved explicit Gamma Stein factors from a maximum-point argument (PDF)
Addendum deriving improved ordinary and weighted Gamma factors; exact constants in the stated small-shape regime.

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